When trying to upgrade from Windows 8.1 to 10, I get a window saying "Something went wrong" with the error code 0x8007001f.

When I press retry, I run through an install while stil in Windows 8.1 but then I get an error on the Windows 10 setup window that says
"Sorry, we're having trouble determining if your PC can run Windows 10. Please close Setup and try again.

Does this have to do with the fact that my USB drive is a removable drive rather than a fixed/local one?

I'm sorry, upgrading Windows when it is booted from a USB drive is not supported, this is a limitation of Microsoft. Our product Windows To Go Upgrader can help upgrade your Windows To Go drive, but it is not free, for more information please refer to:http://www.easyuefi.com/windows-to-go-upgrader/index.html